Fibonacci Sequence
The Fibonacci sequence is a series where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... The ratio of consecutive Fibonacci numbers approaches the golden ratio φ ≈ 1.618.
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Tip: Any three consecutive Fibonacci numbers form a Pythagorean-like identity: F(n)² + F(n)×F(n+1) + F(n+1)² = F(2n+1). Also: the last digit of Fibonacci numbers repeats with period 60 (Pisano period).
- 1F(0) = 0, F(1) = 1
- 2F(n) = F(n-1) + F(n-2)
- 3Closed form (Binet's formula): F(n) = (φⁿ − ψⁿ) / √5
- 4φ = (1+√5)/2 ≈ 1.6180339887 (golden ratio)
F(10)=550,1,1,2,3,5,8,13,21,34,55
F(20)=6,765
F(50)=12,586,269,025About 12.6 billion
| Example | Fibonacci connection |
|---|---|
| Sunflower seeds | 34 and 55 spirals (consecutive Fibonacci) |
| Pineapple scales | 8 and 13 spirals |
| Pinecone bracts | 8 and 13 spirals |
| Rabbit population | Growth model (original 1202 problem) |
| Stock market | Fibonacci retracement levels (38.2%, 61.8%) |
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Fun Fact
The sequence was introduced to Europe by Leonardo of Pisa ('Fibonacci') in 1202 via his book Liber Abaci, through a rabbit breeding problem. However, Indian mathematicians Virahanka and Hemachandra had described the sequence centuries earlier.
References
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